Preheat your oven to 200°C (392°F). Gather your roasting pan and prepare all vegetables by peeling carrots, parsnips, and turnips, then dicing them into 1.5 cm cubes. Rinse and slice the leeks thoroughly to remove any grit.
Spread the diced root vegetables on the roasting pan, drizzle with olive oil, and sprinkle with salt and pepper. Toss well to coat evenly. Roast in the oven for 25-30 minutes, stirring halfway, until vegetables are golden and tender with a sweet, caramelized aroma.
While the vegetables roast, combine the sliced leeks with the chopped thyme and vegetable broth in a large mixing bowl. Mix gently to incorporate the flavors and set aside.
Once the vegetables are done roasting, add them to the bowl with the leek mixture. Toss gently to combine, letting the warm vegetables meld with the herbs and broth, creating a thick, luscious filling.
Roll out the pie crust on a lightly floured surface to about 0.5 cm thick. Fit it into your pie dish, trimming any excess crust around the edges. Use a fork to crimp the edges for a decorative finish.
Spoon the vegetable filling into the prepared crust, spreading it evenly. Brush the edges of the crust lightly with the beaten egg for a shiny finish.
Cover the filling with the second layer of crust or create a lattice pattern if desired. Seal the edges by pressing them together, and cut a small vent in the center to release steam during baking.
Place the assembled pie on a baking sheet to catch any drips, then brush the top with the remaining beaten egg for a golden, glossy look. Bake in the oven for 35-40 minutes, until the crust is deep golden and bubbling at the edges.
Once baked, remove the pie from the oven and let it rest on a wire rack for at least 10 minutes. This allows the filling to set and makes slicing easier without spilling.
Slice the pie carefully with a sharp knife dipped in hot water for smooth cutting. Serve warm, perhaps with a side of greens or a light salad, and enjoy the hearty, caramelized flavors of this rustic dish.
